Definitions

You will be asked to define several of the following terms. The format will be similar to Exams 1 and 2; I will ask you to define a term, and then ask a question intended to see if you understand the definition.

1.

Vector space You do not need to memorize this definition, but you must understand it. If I ask you specific questions that require the complete definition of a vector space, then I will give you the definition on the exam.

2.

The span of $\{{\bf v}_1,{\bf v}_2,\ldots,{\bf v}_k\}$, where ${\bf v}_1,{\bf v}_2,\ldots,{\bf v}_k$ are vectors.

3.

The null space ${\cal N}(A)$ of a matrix A.

4.

The kernel of a linear transformation T.

5.

The column space of a matrix A.

6.

The range of a linear transformation.

7.

Linear independence and linear dependence of a set of vectors.

8.

Linear transformation (in defining linear transformation, you can assume that the meaning of ``transformation'' is known).

9.

$T:V\rightarrow W$ is one-to-one.

10.

$T:V\rightarrow W$ is onto.

11.

Subspace of a vector space.

12.

Basis for a vector space or a subspace.

13.

Coordinates of x with respect to a basis ${\cal B}$

14.

Coordinate mapping, change-of-coordinates matrix

15.

Isomorphism between two vector spaces; two vector spaces are isomorphic.

16.

Dimension of a vector space or subspace; a vector space is finite-dimensional or infinite-dimensional

17.

Rank of a matrix

18.

Eigenvalue, eigenvector of a square matrix.

19.

Characteristic polynomial of a square matrix.

20.

Orthogonal (or orthonormal) basis for a vector space or subspace.

21.

Orthogonal matrix.

22.

Least-squares solution of $A{\bf x}={\bf b}$.